342 Proceedings of Eoyal Society of Edinburgh . [sess. 
On the Expression of any Bordered Skew Determinant 
as a Sum of Products of Pfaffians. By Thomas Muir, 
LL.D. 
(Read December 21, 1896.) 
1. Cayley commences his third paper on Skew Determinants* 
(May, 1854) by recalling his development of them in terms of 
Pfaffians, and then goes on to say : — ■ 
“ J’ai trouve recemment une formule analogue pour le developpe- 
ment d’un determinant gauche horde, tel que 
al234 
a/5 , 
al, 
a2 , 
a3 , 
a4 
1/5, 
11, 
12, 
13, 
14 
2/5, 
21, 
22 
*j_; , 
23, 
24 
3/5, 
31, 
32, 
33, 
34 
4/5, 
41, 
42, 
43, 
44 
Cette formule est : 
al234 >81234 = a/511 . 22 . 33 . 44 
+ a/512 . 12 . 33 . 44 
+ a/513. 13.22.44 
+ a/514. 14.22.33 
+ a/523 . 23 . 11.44 
+ a/524. 24. 11.33 
+ a/534 . 34 . 11 ,'22 
+ a/51234 . 1234 
+ al . /51 . 22 . 33 . 44 
+ a2 . >82 . 11 .33.44 
+ a3 . /53. 11 . 22. 44 
+ a4./54. 11.22.33 
+ al23.>S123. 44 
•+■ al24 . /5124 . 33 
+ al34./5!34. 22 
+ a234.£234. 11 
* Cayley, A., “ Eeclierches ulterieures sur les determinants gauches,” Crelle’s 
Journ., 1. pp. 299-213 ; or Collected Math. Papers, ii. pp. 202-215. 
